This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows on a manifold as well as the author's existing results for orbifolds. It sets the scene for a discussion of Morse Theory on a general proper stack and also paves the way for the categorification of other key aspects of differential geometry such as the tangent bundle and the Lie algebra of vector fields.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
Résumé. Nous définissons un groupoïde de Fréchet-Lie Map(S^1,X) d'ana-foncteurs du cercle vers un groupoïde de Lie X. Ceci fournit une présentation du Hom-champ Hom(S^1,\cX), où \cX est le champ différentiable associé à X. Nous appliquons cette construction au groupoïde de Lie sous-jacent au `gerbe fibré' d'une variété différentiable M; le résultat est un gerbe fibré au-dessus de l'espace des lacets LM de M. Abstract. We define a Fréchet--Lie groupoid Map(S^1,X) of anafunctors from the circle into a Lie groupoid X. This provides a presentation of the Hom-stack Hom(S^1,\cX), where \cX is the di…
In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375], we have discussed the correspondence between two notions of a gerbe over a stack. In [arXiv:2012.08447], we have developed the notion of Chern-Weil map for principal bundles over a Lie groupoid when the Lie groupoid is equipped with a connection. In [arXiv:2012.08442], we discuss the notion of connection on the principal bundle over a Deligne Mumford stack …
We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids.
Stacky Lie groupoids are generalizations of Lie groupoids in which the "space of arrows" of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoids, i.e., actions whose quotients are again differentiable stacks in such a way that the projection onto the quotient is a principal bundle. As an application, we extend the notion of Morita equivalence of Lie groupoids to the realm of stacky Lie groupoids, providing examples tha…
We develop a theory of Lie algebroids over differentiable stacks that extends the standard theory of Lie algebroids over manifolds. In particular we show that Lie algebroids satisfy descent for submersions, define the category of Lie algebroids over a differentiable stack, construct a cohomology theory for these objects, and explain the relation to the theory of $\mathcal{LA}$-groupoids. We construct a number of examples.
We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.
We construct and analyse models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler. We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott's spectral sequence which converges to the cohomology of the classifying space of a Lie group.
We study $S^1$-bundles and $S^1$-gerbes over differentiable stacks in terms of Lie groupoids, and construct Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvature.
We introduce and study measures and densities (= geometric measures) on differentiable stacks, using a rather straightforward generalization of Haefliger's approach to leaf spaces and to transverse measures for foliations. In general we prove Morita invariance, a Stokes formula which provides reinterpretations in terms of (Ruelle-Sullivan type) algebroid currents, and a Van Est isomorphism. In the proper case we reduce the theory to classical (Radon) measures on the underlying space, we provide explicit (Weyl-type) formulas that shed light on Weinstein's notion of volumes of differentiable sta…
Let $\mathcal{G}$ be a Lie groupoid. The category $B\mathcal{G}$ of principal $\mathcal{G}$-bundles defines a differentiable stack. On the other hand, given a differentiable stack $\mathcal{D}$, there exists a Lie groupoid $\mathcal{H}$ such that $B\mathcal{H}$ is isomorphic to $\mathcal{D}$. Define a gerbe over a stack as a morphism of stacks $F\colon \mathcal{D}\rightarrow \mathcal{C}$, such that $F$ and the diagonal map $Δ_F\colon \mathcal{D}\rightarrow \mathcal{D}\times_{\mathcal{C}}\mathcal{D}$ are epimorphisms. This paper explores the relationship between a gerbe defined above and a Mori…
The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of $(+1)$ Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests: (1) We introduce a $\mathbb Z$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its ho…
This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for stacks. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids. Differentiable stacks are geometric objects which have manifolds and orbifolds as special instances, and can be presented as the transverse geometry of a Lie groupoid. Two Lie groupoids are equivalent if they are presenting the same stack, and proper groupoids are presentations of separated stacks, which by the linearization theorem are locally modeled by l…
We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.
In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds, and relate it to the covering theory of stacks. As applications, we obtain a geometric version of Rieffel's theorem on irrational rotations of the circle, we compute the stack-theoretic fundame…
For a compact manifold M and a differentiable stack \cX presented by a Lie groupoid X, we show the Hom-stack Hom(M,\cX) is presented by a Fréchet-Lie groupoid Map(M,X) and so is an infinite-dimensional differentiable stack. We further show that if \cX is an orbifold, presented by a proper étale Lie groupoid, then Map(M,X) is proper étale and so presents an infinite-dimensional orbifold.
We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connections and curvings for groupoid $S^1$-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for $S^1$-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms …
We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the groupoid. Sections of a square root of this line bundle …
In this paper, we develop twisted $K$-theory for stacks, where the twisted class is given by an $S^1$-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure $K^i_α\otimes K^j_β\to K^{i+j}_{α+β}$ are derived. Our approach provides a uniform framework for studying various twisted $K$-theories including the usual twisted $K$-theory of topological spaces, twisted equivariant $K$-theory, and the twisted $K$-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted $K$-groups can be ex…
In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. With…
A new cohomology, induced by a vector field, is defined on pairs of differential forms ($1$--differentiable forms) in a manifold. It is proved a link with the classical de Rham cohomology and an $1$-differentable cohomology of Lichnerowicz type associated to an one form. Also, the case when the manifold is complex and the vector field is holomorphic is studied. Finally, an application of this theory to the harmonicity of $1$-differentiable forms is studied in a particular case.
We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlas-independent and provides a bijection between isomorphism classes of principal circle bundles (with or without connections) and second cohomology groups of certain chain complexes.
We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid $G$-extensions, which we call "connections on gerbes", and study the induced connections on various associated bundles. We also prove analogues of the Bianchi identities. In particular, we develop a cohomology theory which measures the existence of connections and curvings for $G$-gerbes over stacks. We also introduce $G$-central extensions of groupoids, generalizing the standard groupoid $S^1$-central extensions. As an example, we apply…
We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the geometric stack. We show that the category of vector fields on the Lie groupoid is equivalent to the category of vector fields on the stack. The category of vector fields on the Lie groupoid has a Lie 2-algebra structure built from known (ordinary) Lie brackets on multiplicative vector fields of Mackenzie and Xu and the global sections of the Lie algebroid of the Lie groupoid. After giving a precise…
We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack …
In this work we introduce the category of multiplicative sections of an $\la$-groupoid. We prove that this category carries natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including…
We present some features of the smooth structure, and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures - it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek's framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical str…
We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is continuously reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differenial forms, which statisfies Smith's version of de Rham's theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.
We show how an affine connection on a Riemannian manifold occurs naturally as a cochain in the complex for Leibniz cohomology of vector fields with coefficients in the adjoint representation. The Leibniz coboundary of the Levi-Civita connection can be expressed as a sum of two terms, one the Laplace-Beltrami operator and the other a Ricci curvature term. The vanishing of this coboundary has an interpretation in terms of eigenfunctions of the Laplacian. Additionally, we compute the Leibniz cohomology with adjoint coefficients for a certain family of vector fields on Euclidean ${\bf{R}}^n$ corre…
We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an étale topological or differentiable stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of 2-categories between small stacks over an étale stack and local homeomorphisms over it. We go on to characterize small sheaves and gerbes. We show that ineffective data of étale stacks is completely described by the theory of small gerbes. Furthermore, it is shown that étale stacks (and in…
In this sequel to works D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we re-examine --- and reformulate when in need --- several basic notions in super $C^{\infty}$-algebraic geometry as guided by the mathematical formulation of Ramond-Neveu-Schwarz fermionic strings and of Green-Schwarz fermionic strings from the viewpoint of Grothendieck on Algebraic Geometry. Two theorems that are the super counterpart of Theorem~3.1.1 and Theorem~3.2.1 of D(11.3.1) are proved. They unify the notion of "smooth maps from an Azumaya/matrix…
We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stacks. We demonstrate that the resulting theory generalizes the parametrized cobordism of Galatius-Madsen-Tillman-Weiss. The theory has the feature of being both versatile and general, allowing for the inclusion of families of various geometric data, such as metrics on manifolds and…
